This tag is for questions about the foundations of mathematics, and the formalization of mathematical concepts in foundational theories (e.g. set theory, category theory, and type theory).

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What is the physical significance of arithmetic operations?

Here is an example of what I mean by physical significance: When we use some geometric or trigonometric identity, let us say Pythagoras' theorem to calculate the length of the diagonal of a field, ...
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Guidelines on how to formalize mathematics from its foundations.

Let's say I wanted to understand mathematics in the most formally rigoruous way, meaning from the basic understanding of arithmetic, I'd want every operation I make to have a formal proof, and ...
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63 views

How to construct $\mathbb{R}^N$ where $N$ is a random variable?

How does one rigorously construct $\mathbb{R}^N$ where $N$ is a $\mathbb{Z}^{++}$-valued random variable on some Borel probability space $(\Omega,\mathcal{B},\mathbb{P})$? Would someone be so kind ...
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1answer
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Does the foundations of Arithmetic need to be effective?

I was reading about Godel's incompleteness theorem which is true for any formal theory that satisfies certain properties. One of these properties is the following: "The theory is assumed to be ...
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3answers
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Motivation for different mathematics foundations

I've been studying an introductory book on set theory that uses the ZFC set of axioms, and it's been really exciting to construct and define numbers and operations in terms of sets. But now I've seen ...
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1answer
91 views

A few questions about a true but unprovable statement

Can someone explain to me what this comment means: If ZFC is not a sound theory, a true but unprovable statement may be refutable and therefore decidable. What is a sound theory? What is ...
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94 views

Is there a universal way to define cartesian product with arbitrary many terms?

Suppose we are working with some set theory where primitives are sets and membership. Starting from that, we can give a prescription to define what it means to be an ordered pairs $(a,b)$. This allows ...
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1answer
35 views

By W&R's type theory, has Russell abandoned his view in his Principles §7?

The Principles. §7 : Thus, for example, the proposition "x and y are numbers implies $(x+y)^2 = x^2 + 2xy + y^2$ " still holds equally if for x and y we substitute Socrates and Plato[2]: both ...
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1answer
109 views

What problems remain unanswered today regarding the nature of variables?

So far, to me, the greatest difficulty in studying philosophy is to recognize the importance of the problems: Exactly what make philosophers think these problems are worthy subject of study? Take ...
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67 views

Extension of the definitions of composition of total and partial functio

In a comment to the question: definition of the domain of composite partial functions , user theGuest asked: “can we define g∘f more generally,i.e when f:X⇁Y and g:W⇁Z ?” Could the following ...
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3answers
184 views

Can $1=0$ ever make sense?

Can $1=0$ ever make sense? In more detail, under which interpretation of $0$ and $1$ is $1=0$ possible? what are the consequences of such a result? Under which interpretations would $1=0$ never be ...
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1answer
39 views

Definition Of Composition Of Functions

The standard definition of function composition is: Let $f : A \to B$, $g : B \to C$ then there is a composite function $g \circ f : A \to C$, given by $(g \circ f)(a) = g(f(a))$ with $a \in A$ Why ...
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0answers
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What is the Best Introduction to Dedekind Cuts?

I'm looking for a clear, thorough, and easy-to-follow introduction to Dedekind cuts that is specifically geared towards those with an interest in foundational issues. So far, the discussions that I ...
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1answer
83 views

Is there any problem that is proved not independent of ZFC but the problem itself is not proved yet?

Is there any problem in mathematics that is proved not independent of ZFC but the problem itself is not proved yet?
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1answer
63 views

Proving a theorem of logic

At the moment I'm going through a book which treats logic in a very rigorous axiomatic way. But I just got stuck in this theorem that I can't seem to be able to solve (I'm still trying hard). The ...
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5answers
443 views

Foundation of ordering of real numbers

This might be a silly question, but what is the mathematical foundation for the ordering of the real numbers? How do we know that $1<2$ or $300<1000$... Are the real numbers simply defined as ...
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3answers
70 views

Give a serious explanation of the difference between an equation and a function.

What's the difference between an equation and a function? I mean, I am not seeking for a high school-like answer like "an equation has an equals sign". I want to know what is the fundamental ...
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4answers
164 views

mathematical proof vs. first-order logic deductions

For a long time I thought that the standard mathematical proofs, only were an informal or imperfect way of writing the proof in the language of first-order logic. When I say standard mathematical ...
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42 views

References for the foundations of mathematics

I´m a physicist and I´d like to learn about deep questions on the foundations of mathematics. I´m looking for a acessible introduction written in a language a physicist could understand, and then ...
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1answer
38 views

Problems with this Cartesian Product definition

Supposed I do not define ordered pair in the usual Kuratowski way $(x,y) = \{\{x\},\{x,y\}\}$. I left the ordered pair undefined but with the propriety $(x,y) = (x',y') \iff x=x'\text{ and }y=y'$. ...
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0answers
106 views

Is this a way to construct mathematics?(logic vs. set theory)

I recently asked a question about the fact that logic and set theory seems circular. link I got a lot of good and thoughtful answers, that probably explains everything, but I must admit I did not ...
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9answers
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Does mathematics become circular at the bottom? What is at the bottom of mathematics? [duplicate]

I am trying to understand what mathematics is really built up of. I thought mathematical logic was the foundation of everything. But from reading a book in mathematical logic, they use ...
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1answer
36 views

A foundation where objects are unlabelled

Okay, this may seem a crazy question. If it does, it's because I've been thinking too much about foundations, recently. I've read a bit on ZFC, on the need for universes for categories, on HoTT ...
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2answers
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How can we work with field extensions when our base fields aren't actually subfields?

I've been wondering this for a little while. Say we are working with the rational numbers $\mathbb{Q}$, and then we wish to talk about the extension fields $E$ of $\mathbb{Q}$, by which we mean the ...
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0answers
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Defn of invariant SET under an equivalence relation?

This is, I am sure, an easy question but something I did not see in my undergrad years and wikipedia does not define it. For a function or a relation or a property to be invariant under an equivalence ...
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11answers
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What are some results that shook the foundations of one or more fields of mathematics? [closed]

An example would be the proof that $\sqrt{2}$ is not rational, which was a violation of some fundamental assumptions that mathematicians at the time made about numbers. Another would be Russell's ...
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What is the likely future of Univalent Foundations?

Univalent foundations has been hyped up as the foundation for mathematics for the future in articles such as this one. Now I've given HoTT a brief look, and at least seen that it appears on the face ...
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5answers
874 views

What do people mean by “finite”?

Many arguments about the foundations or philosophy of mathematics centre on the question of whether or not there exist objects or entities (such as certain sets) which are not "finite". (For ...
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3answers
166 views

Lists of sets as objects of ZF axiomatics

I have a naive question about foundations of mathematics. A common opinion of most mathematicians is that the essential part of mathematics can be reduced to ZF(C) axioms. I do not quite understand ...
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5answers
1k views

Are there any nontrivial examples of contradictions arising in non-foundational or applied math due to naive set theory?

I understand that naive set theory, whose axioms are extensionality and unrestricted comprehension, is inconsistent, due to paradoxes like Russell, Curry, Cantor, and Burali-Forti. But these all ...
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1answer
74 views

Categorical Foundations text

I've heard that someone's thought up a way of using category theory, involving something called topoi, as a foundation for mathematics. If this is true then are there any texts which explain such a ...
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2answers
83 views

When can variables simply be variables?

This may seem a somewhat strange question, but I've been tying myself in knots about it recently. When constructing a polynomial ring, you must formally define a polynomial as an ordered ω-tuple, ...
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2answers
301 views

Why is it important, that mathematics can be formalized in set theory?

Why is it important, that mathematics can be formalized in set theory? As one can read in the thread Are there areas of mathematics that cannot be formalized in set theory? Today known mathematical ...
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3answers
280 views

Why is something not a field if it's a proper class?

Why is it a convention to say that, for example, the nimbers and surreal numbers aren't fields because they don't form sets? Is this just pedantry?
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2answers
61 views

The class of all functions between classes (NBG)

Is it possible in NBG (von Neumann-Bernays-Gödel set theory) to construct the class of all functions $X \to Y$ between two (proper) classes $X,Y$? I guess that this does not work. In the special case ...
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2answers
62 views

Construction of Natural Numbers

I am trying to prove that the natural numbers can be constructed from the product of a power of $2$ and an odd number. For all $n \neq 0$ in the natural numbers, $n = (2k+1)(2^p)$, where $k$ and $p$ ...
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1answer
131 views

Definiton of Limit and Foundational problems.

I am new to Category theory and I have a quite strong foundational problem. For example, let's start from the definition of Limit stated by wikipedia ...
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1answer
185 views

I need help understanding Frege's definition of number

I have really been trying to understand Frege's definition of a number or at least gain a strong intuition of it. However, my attempts have not been fruitful. If someone could help me it would be much ...
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1answer
56 views

Defining integer sum without using infinite sets

In ZFC minus infinity (let us call this system $T$), one can still define ordinals, and then define integers as ordinals all of whose members are zero or successor ordinals. Combining the power set ...
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1answer
89 views

Integer induction without infinity

In ZFC minus infinity (let us call this T), one can still define ordinals, and then define integers as ordinals all of whose members are zero or successor ordinals. I am looking for a formula $\psi$ ...
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2answers
48 views

how can we express finiteness as a first order property?

I don't know much about set theory but I read that in ZFC a set is finite when there are no bijections from the set to a proper subset of itself. It seems to me however that quantifying over subsets ...
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2answers
101 views

Function, Relation, Operation and Cartesian Product

An operation is a kind of function. A function is a kind of relation. A relation is a subset of a Cartesian product. A Cartesian product is an operation. Back to 1. It seems to me that there's ...
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1answer
115 views

Why does 2+2 equal to 4? [duplicate]

The question is in the title. I am very appreciative of any time and concern put into belaboring this relatively little problem.
57
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9answers
4k views

Why is the construction of the real numbers important?

There are a lot of books, specially in Real Analysis and set theory, which define the real numbers by Cauchy sequences or Dedekind cuts. So my question is why don't we simply define the Real numbers ...
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1answer
39 views

Can boolean logic compute any sort of mathematical operation?

Computers fundamentally do logical operations on the input and memory they have (as far as I know). Computers are used by mathematicians to do all sorts of mathsy operations (as far as I know). Does ...
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0answers
144 views

How should one understand the foundation of set theory?

I have read the answer of Carl Mummert for the question on how to avoid circularity. I would like to ask further as I want to study models of set theory. As I understand, with say assuming the ...
3
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1answer
129 views

Does internalization loses informations everywhere?

It is well known that a group object in Grp is necessarily abelian. This can be understood as "internalization loses information". Indeed, if one was to study group theory by looking at group objects ...
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1answer
57 views

A nonempty class of isomorphic groups defines a group

The context of this question is from the definition of the sporadic Mathieu group $M_{23}$, which (in one possible definition) is the stabilizer of a point in $M_{24}$, which is a certain subgroup of ...
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2answers
239 views

What is so special about the real and complex numbers?

When I was studying linear algebra, the first thing we were introduced was the idea of fields. In studying analysis (and when studying inner product spaces etc), we restricted our possible fields to ...
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3answers
61 views

Abstracting objects

I understand that one way of defining a mathematical object such as a group is to take an object we already know to exist, for example the integers, and take away some properties from them. This is ...